Time-dependent Traveling Salesman Problem Research Articles

Hybrid optimization methods for time-dependent sequencing problems

In this paper, we introduce novel optimization methods for sequencing problems in which the setup times between a pair of tasks depend on the relative position of the tasks in the ordering. Our proposed methods rely on a hybrid approach where a constraint programming model is enhanced with two distinct relaxations: One discrete relaxation based on multivalued decision diagrams, and one continuous relaxation based on linear programming. Both relaxations are used to generate bounds and enhance constraint propagation. Experiments conducted on three variants of the time-dependent traveling salesman problem indicate that our techniques substantially outperform general-purpose methods, such as mixed-integer linear programming and constraint programming models.

The Time Dependent Traveling Salesman Planning Problem in Controlled Airspace

The integration of drones into civil airspace is one of the most challenging problems for the automation of the controlled airspace, and the optimization of the drone route is a key step for this process. In this paper, we optimize the route planning of a drone mission that consists of departing from an airport, flying over a set of mission way points and coming back to the initial airport. We assume that during the mission a set of piloted aircraft flies in the same airspace and thus the cost of the drone route depends on the air traffic and on the avoidance maneuvers used to prevent possible conflicts. Two air traffic management techniques, i.e., routing and holding, are modeled in order to maintain a minimum separation between the drone and the piloted aircraft. The considered problem, called the Time Dependent Traveling Salesman Planning Problem in Controlled Airspace (TDTSPPCA), relates to the drone route planning phase and aims to minimize the total operational cost. Two heuristic algorithms are proposed for the solution of the problem. A mathematical formulation based on a particular version of the Time Dependent Traveling Salesman Problem, which allows holdings at mission way points, and a Branch and Cut algorithm are proposed for solving the TDTSPPCA to optimality. An additional formulation, based on a Travelling Salesman Problem variant that uses specific penalties to model the holding times, is proposed and a Cutting Plane algorithm is designed. Finally, computational experiments on real-world air traffic data from Milano Linate Terminal Maneuvering Area are reported to evaluate the performance of the proposed formulations and of the heuristic algorithms.

Lagrangean relaxation approach to joint optimization for production planning and scheduling of synchronous assembly lines

This paper focuses on simultaneous optimisation of production planning and scheduling problem over a time period for synchronous assembly lines. Differing from traditional top-down approaches, a mixed integer programming model which jointly considers production planning and detailed scheduling constraints is formulated, and a Lagrangian relaxation method is developed for the proposed model, whereby the integrated problem is decomposed into planning, batch sequencing, tardiness and earliness sub-problems. The scheduling sub-problem is modelled as a time-dependent travelling salesman problem, which is solved using a dynasearch algorithm. A proposition of Lagrangian multipliers is established to accelerate the convergence speed of the proposed algorithm. The average direction strategy is employed to solve the Lagrangian dual problem. Test results demonstrate that the proposed model and algorithm are effective and efficient.

Analysis and Branch-and-Cut Algorithm for the Time-Dependent Travelling Salesman Problem

Given a graph whose arc traversal times vary over time, the time-dependent travelling salesman problem (TDTSP) consists in finding a Hamiltonian tour of least total duration covering the vertices of the graph. The contribution of this paper is twofold. First, we describe a lower and upper bounding procedure that requires the solution of a simpler (yet NP-hard) asymmetric travelling salesman problem (ATSP). In addition, we prove that this ATSP solution is optimal for the TDTSP if all the arcs share a common congestion pattern. Second, we formulate the TDTSP as an integer linear programming model for which valid inequalities are devised. These inequalities are then embedded into a branch-and-cut algorithm that is able to solve instances with up to 40 vertices.

Facets and valid inequalities for the time-dependent travelling salesman problem

The Time-Dependent Travelling Salesman Problem (TDTSP) is a generalization of the traditional TSP where the travel cost between two cities depends on the moment of the day the arc is travelled. In this paper, we focus on the case where the travel time between two cities depends not only on the distance between them, but also on the position of the arc in the tour. We consider two formulations proposed in the literature, we analyze the relationship between them and derive several families of valid inequalities and facets. In addition to their theoretical properties, they prove to be very effective in the context of a Branch and Cut algorithm.

Biobjective planning of an active debris removal mission

The growth of the orbital debris population has been a concern to the international space community for several years. Recent studies have shown that the debris environment in Low Earth Orbit (LEO, defined as the region up to 2000km altitude) has reached a point where the debris population will continue to increase even if all future launches are suspended. As the orbits of these objects often overlap the trajectories of satellites, debris create a potential collision risk. However, several studies show that about 5 objects per year should be removed in order to keep the future LEO environment stable. In this article, we propose a biobjective time dependent traveling salesman problem (BiTDTSP) model for the problem of optimally removing debris and use a branch and bound approach to deal with it.

Machine scheduling in the presence of sequence-dependent setup times and a rate-modifying activity

Rate modifying activity (RM) is a type of maintenance after which the processing rate of the machine increases. RM is a very new topic in academic studies. However, it is very common in real world situations. In this paper, we study the integrated problem of assigning a common due-date to all jobs, scheduling the jobs and making decisions about the position of RM in a single machine environment in which the setup times are sequence dependent. The objective is minimising the summation of earliness costs, tardiness costs and due date related costs. This problem has never been studied in the literature with any arbitrary criterion. We construct a time-dependent travelling salesman problem (TDTSP) formulation for this problem. The position of the optimal common due date and some dominance properties for the position of RM are presented. A branch and bound (B&B) procedure is developed to solve the problem optimally. Numerical results justify the effectiveness of the B&B method for small problems. For larger problems, two robust metaheuristics are proposed.

The time dependent traveling salesman problem: polyhedra and algorithm

The time dependent traveling salesman problem (TDTSP) is a generalization of the classical traveling salesman problem (TSP), where arc costs depend on their position in the tour with respect to the source node. While TSP instances with thousands of vertices can be solved routinely, there are very challenging TDTSP instances with less than 100 vertices. In this work, we study the polytope associated to the TDTSP formulation by Picard and Queyranne, which can be viewed as an extended formulation of the TSP. We determine the dimension of the TDTSP polytope and identify several families of facet-defining cuts. We obtain good computational results with a branch-cut-and-price algorithm using the new cuts, solving almost all instances from the TSPLIB with up to 107 vertices.

Two improved formulations for the minimum latency problem

In network problems, latency is associated with the metric used to evaluate the length of the path from a root vertex to each vertex in the network. In this work we are dealing with two applications or variations of the minimum latency problem known as the repairman problem and the deliveryman problem. We have developed two integer formulations for the minimum latency problem and compared them with other two formulations from the literature for the time-dependent traveling salesman problem. The present work highlights the similarities and differences between the different formulations. In addition, we discuss the convenience of including a set of constraints in order to reduce the computation time needed to reach the optimal solution. We have carried out extensive computational experimentation on asymmetrical instances, since they provide the characteristics of the deliveryman and repairman problems in a better way.

A pattern recognition lexi search approach to generalized time-dependent travelling salesman problem

In this paper the “Generalized Time-dependent Travelling Salesman Problem” is a three dimensional travelling salesman problem where the cost matrix C (i, j, k) is the cost of the salesman visiting from city i to city j at time (facility) k. The cost matrix C(i,j,k) [i,j = 1,2,3,…..,n;k = 1,2,3,…..,m] is given. There are n cities and N = {1, 2, 3,…, n}. We are given a partition of cities into groups and we require to find a minimum length tour that includes at least one city from each group. The problem is to find a feasible tour for m(<n) cities for the salesman with minimum total cost with given number of cities from each group with the restriction that he visits only one pair of cities at each point of time(facility) i.e., at a particular time he has to travel from one city to another city. We propose a Lexi Search algorithm based on Pattern Recognition Technique for solving “Generalized Time-dependent Traveling Salesman Problem” with an illustrative example. For this problem a computer program is developed for the algorithm and is tested. It is observed that it takes less time for solving fairly higher dimension problem also.

An optimisation-based seasonal sugarcane harvest scheduling decision support system for commercial growers in South Africa

An ongoing sugarcane decision support research project in South Africa is aimed at developing a decision support system capable of providing computerised support to those charged with the task of scheduling sugarcane harvesting operations in South Africa. In situations where the number of fields is large and when the conditions under which the crops are growing change frequently—for example as a result of climatic, biological or management-related events—computerised support is applicable. Commercial growers have provided data suitable for regression modelling of the parameters that govern the values and costs involved, and have participated in two consecutive preliminary system evaluation and development experiments conducted during the 2009 and 2010 harvesting seasons. The optimisation models underlying the decision support system are based on a time-dependent travelling salesman problem formulation and are solved approximately by means of a tabu search in a Microsoft Visual Basic for Applications (VBA) for Excel environment. The growers who participated in the evaluation experiments responded positively to the decision support system and stated that it may be useful to large-scale sugarcane producers as well as emerging growers. The authors’ findings are that the decision support system provides support in practical sugarcane harvest scheduling and that one series of regression fits is required for each agroclimatically and management-wise homogenous area.

Optimization of an Occulter-Based Extrasolar-Planet-Imaging Mission

Anovel approach to extrasolar-planet imaging uses a pair of satellites: a telescope and an occulter, where the latter is placed in the line of sight between the telescope and the star system to be imaged in order to enhance the telescope’s imaging capability. The optimal configuration of this satellite formation around sun–Earth L2 halo orbits is studied. Trajectory optimization of the occultermotion between imaging sessions of different stars is performed using a range of different criteria and methods. Thus, the global optimization problem is transformed into a time-dependent traveling salesman problem. The time-dependent traveling salesman problem is augmentedwith various constraints that arrive from the mission, and this problem is solved by employing simulated annealing and branching algorithms. For a concrete understanding of the feasibility of themission, the performance of an example spacecraft, the Small Missions for Advanced Research in Technology (SMART-1), is analyzed.

Natural and extended formulations for the Time-Dependent Traveling Salesman Problem

In this paper, we present a new formulation for the Time-Dependent Traveling Salesman Problem (TDTSP). We start by reviewing well known natural formulations with some emphasis on the formulation by Picard and Queyranne (1978) [22]. The main feature of this formulation is that it uses, as a subproblem, an exact description of the n-circuit problem. Then, we present a new formulation that uses more variables and is based on using, for each node, a stronger subproblem, namely an n-circuit subproblem with the additional constraint that the corresponding node is not repeated in the circuit. Although the new model has more variables and constraints than the original PQ model, the results given from our computational experiments show that the linear programming relaxation of the new model gives, for many of the instances tested, gaps that are close to zero. Thus, the new model is worth investigating for solving TDTSP instances. We have also provided a complete characterization of the feasible set of the corresponding linear programming relaxation in the space of the variables of the PQ model. This characterization permits us to suggest alternative methods of using the proposed formulations.

An integer programming approach for the time-dependent TSP

The Time-Dependent Travelling Salesman Problem (TDTSP) is a generalization of the traditional TSP where the travel cost between two cities depends on the moment of the day the arc is travelled. In this paper, we focus on the case where the travel time between two cities depends not only on the distance between them, but also on the position of the arc in the tour. We consider the formulations proposed in Picard and Queryanne [J.C. Picard and M. Queyranne. The time-dependent traveling salesman problem and its application to the tardiness problem in one-machine scheduling. Operations Res., 26(1):86–110, 1978] and Vander Wiel and Sahinidis [R.J. Vander Wiel and N.V. Sahinidis. An exact solution approach for the time-dependent traveling-salesman problem. Naval Res. Logist., 43(6):797–820, 1996], analyze the relationship between them and derive some valid inequalities and facets. Computational results are also presented for a Branch and Cut algorithm (B&C) that uses these inequalities, which showed to be very effective.

The time-dependent traveling salesman problem and single machine scheduling problems with sequence dependent setup times

In this paper, we consider scheduling problems on a single machine in a sequence dependent setup environment. For these problems, we introduce several integer programming formulations of varying size and strength. Computational experiments conducted on instances of 1 | s i j | ∑ C j (i.e. minimizing total flow time on a single machine under sequence dependent setup times) and 1 | s i j | ∑ T j (i.e. minimizing total tardiness on a single machine under sequence dependent setup times) illustrate the benefits of using stronger formulations, even though the computation of their LP relaxation is more time consuming. Incorporating these improved LP relaxation bounds, we propose an exact branch-and-bound algorithm able to solve instances of 1 | s i j | ∑ C j and 1 | s i j | ∑ T j having up to 50 and 45 jobs respectively.

A branch-and-cut algorithm for a production scheduling problem with sequence-dependent and time-dependent setup times

This paper introduces and compares three different formulations of a production scheduling problem with sequence-dependent and time-dependent setup times on a single machine. The setup is divided into two parts: one that can be performed at any time and another one that is restricted to be performed outside of a given time interval. As a result, the setup time between two jobs is a function of the completion time of the first job. The problem can be formulated as a time-dependent traveling salesman problem, where the travel time between two nodes is a function of the departure time from the first node. We show that the resulting formulation can be strengthened to provide better linear programming relaxation lower bounds. We also introduce several families of valid inequalities which are used within a branch-and-cut algorithm. Computational experiments show that this algorithm can solve some instances with up to 50 jobs within reasonable computing times.

Time-Dependent Travelling Salesman Problem

In this paper the Time-Dependent Travelling Sales Man Problem is a 3-dimensional travelling sales man problem where the cost matrix C(i, j, k) is the cost of the sales man visiting from city i to city j at time (facility) k. The cost matrix C(i, j,k) (i, j, k = 1,2, — n) is given. The problem is to find a tour for n cities for the travelling sales man with the restriction that he visits only one pair of cities at each point of time (facility), i.e. at a particular time he has to travel from one city to another city only. The objective is to find the feasible tour to the sales man such that the total cost is minimum. For this problem a computer programme is developed for the algorithm and is tested. It is observed that it takes less time for solving fairly higher dimension problems also. For this problem the definition, notation and algorithm are illustrated with the help of a numerical example.

Total flowtime and makespan for a no-wait m-machine flowshop with set-up times separated

This paper addresses the m-machine no-wait flowshop problem where the set-up time of a job is separated from its processing time. The performance measures considered are the total flowtime and makespan. The scheduling problem for makespan reduces to the travelling salesman problem (TSP), and the scheduling problem for total flowtime reduces to the time-dependent travelling salesman problem (TDTSP). Non-polynomial time solution methods are presented, along with a polynomial heuristic.

Online and Offline Algorithms for the Time-Dependent TSP with Time Zones

The time-dependent traveling salesman problem (TDTSP) is a variant of TSP with time-dependent edge costs. We study some restrictions of TDTSP where the number of edge cost changes are limited. We find competitive ratios for online versions of TDTSP. From these we derive polynomial time approximation algorithms for graphs with edge costs one and two. In addition, we present an approximation algorithm for the orienteering problem with edge costs one and two.

The time-dependent traveling salesman problem

A problem often considered in Operations Research and Computational Physics is the traveling salesman problem, in which a traveling salesperson has to find the shortest closed tour between a given set of cities touching each city exactly once. The distances between the single nodes are known to the traveling salesperson. An extension of this problem is the time-dependent traveling salesman problem, in which these distances vary in time. I will show how this more complex problem is treated with physical optimization algorithms like simulated annealing. I will present results for the problem of the 127 beergardens in the area of Augsburg, in which I define a traffic zone in which traffic jams occur in the afternoon.